chapter 10 comparisons involving means
DESCRIPTION
Chapter 10 Comparisons Involving Means. Inferences About the Difference Between Two Population Means: when s 1 and s 2 Known. Inferences About the Difference Between Two Population Means: when s 1 and s 2 Unknown. Inferences About the Difference Between - PowerPoint PPT PresentationTRANSCRIPT
1 © 2009 Econ-2030(Dr. Tadesse)
Chapter 10 Comparisons Involving
Means Inferences About the Difference Between
Two Population Means: when s 1 and s 2 Known
Inferences About the Difference Between Two Population Means: Matched Samples
Inferences About the Difference Between Two Population Means: when s 1 and s 2 Unknown
Introduction to Analysis of Variance (Inference about the difference between more than two population means
2 © 2009 Econ-2030(Dr. Tadesse)
Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Known
Interval Estimation of m 1 – m 2
Hypothesis Tests About m 1 – m 2
3 © 2009 Econ-2030(Dr. Tadesse)
Estimating the Difference BetweenTwo Population Means
Let m1 equal the mean of population 1 and m2 equal
the mean of population 2. The difference between the two population means is m1 - m2. To estimate m1 - m2, we will select a simple random
sample of size n1 from population 1 and a simple
random sample of size n2 from population 2. Let equal the mean of sample 1 and
equal the mean of sample 2.
x1 x2
The point estimator of the difference between the
means of the populations 1 and 2 is .x x1 2
4 © 2009 Econ-2030(Dr. Tadesse)
Expected Value
Sampling Distribution of x x1 2
E x x( )1 2 1 2 m m
Standard Deviation (Standard Error)
s s sx x n n1 2
12
1
22
2
where: s1 = standard deviation of population 1 s2 = standard deviation of population 2
n1 = sample size from population 1 n2 = sample size from population 2
5 © 2009 Econ-2030(Dr. Tadesse)
Interval Estimate
Interval Estimation of m1 - m2: s 1 and s 2 Known
2 21 2
1 2 / 21 2
x x zn ns s
where: 1 - is the confidence coefficient
6 © 2009 Econ-2030(Dr. Tadesse)
Example:
Interval Estimation of m1 - m2: s 1 and s 2 Known
In a test of driving distance using a mechanicaldriving device, a sample of Par golf balls wascompared with a sample of golf balls made by Rap,Ltd., a competitor. The sample statistics appear on thenext slide.
Par, Inc. is a manufacturerof golf equipment and hasdeveloped a new golf ballthat has been designed toprovide “extra distance.”
7 © 2009 Econ-2030(Dr. Tadesse)
Interval Estimation of m1 - m2: s 1 and s 2 Known
Sample SizeSample Mean
Sample #1Par, Inc.
Sample #2Rap, Ltd.
120 balls 80 balls275 yards 258 yards
Based on data from previous driving distancetests, the two population standard deviations areknown with s 1 = 15 yards and s 2 = 20 yards.
8 © 2009 Econ-2030(Dr. Tadesse)
Interval Estimation of m1 - m2: s 1 and s 2 Known
Develop a 95% confidence interval estimate of the difference between the mean driving distances ofthe two brands of golf ball.
9 © 2009 Econ-2030(Dr. Tadesse)
Estimating the Difference BetweenTwo Population Means
m1 – m2 = difference between the mean distances
x1 - x2 = Point Estimate of m1 – m2
Population 1Par, Inc. Golf Ballsm1 = mean driving
distance of Pargolf balls
Population 2Rap, Ltd. Golf Ballsm2 = mean driving
distance of Rapgolf balls
Simple random sample of n2 Rap golf ballsx2 = sample mean distance for the Rap golf balls
Simple random sample of n1 Par golf ballsx1 = sample mean distance for the Par golf balls
10 © 2009 Econ-2030(Dr. Tadesse)
Point Estimate of m1 - m2
Point estimate of m1 m2 = x x1 2
where:m1 = mean distance for the population of Par, Inc. golf ballsm2 = mean distance for the population of Rap, Ltd. golf balls
= 275 258
= 17 yards
11 © 2009 Econ-2030(Dr. Tadesse)
x x zn n1 2 2
12
1
22
2
2 217 1 96 15
1202080
s s
/ . ( ) ( )
Interval Estimation of m1 - m2:s 1 and s 2 Known
We are 95% confident that the difference betweenthe mean driving distances of Par, Inc. balls and Rap,Ltd. balls is 11.86 to 22.14 yards.
17 + 5.14 or 11.86 yards to 22.14 yards
12 © 2009 Econ-2030(Dr. Tadesse)
Hypothesis Tests About m 1 m 2:s 1 and s 2 Known
Hypothesis Testing
1 2 02 21 2
1 2
( )x x Dz
n ns s
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 0 1 2 0: H D
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 1 2 0: aH D
Left-tailed Right-tailed Two-tailed Test Statistic
13 © 2009 Econ-2030(Dr. Tadesse)
Hypothesis Tests About m 1 m 2:s 1 and s 2 Known
Can we conclude, using = .01, that the mean drivingdistance of Par, Inc. golf ballsis greater than the mean drivingdistance of Rap, Ltd. golf balls?
14 © 2009 Econ-2030(Dr. Tadesse)
H0: m1 - m2 < 0
Ha: m1 - m2 > 0
where: m1 = mean distance for the population of Par, Inc. golf balls m2 = mean distance for the population of Rap, Ltd. golf balls
1. Develop the hypotheses.
Hypothesis Tests About m 1 m 2:s 1 and s 2 Known
2. Specify the level of significance. = .01
15 © 2009 Econ-2030(Dr. Tadesse)
3. Compute the value of the test statistic.
Hypothesis Tests About m 1 m 2:s 1 and s 2 Known
s s
1 2 02 21 2
1 2
( )x x Dz
n n
2 2(235 218) 0 17 6.492.62(15) (20)
120 80
z
16 © 2009 Econ-2030(Dr. Tadesse)
Hypothesis Tests About m 1 m 2:s 1 and s 2 Known
5. Compare the Test Statistic with the Critical Value.Because z = 6.49 > 2.33, we reject H0.
Using the Critical Value Approach
For = .01, z.01 = 2.334. Determine the critical value and rejection rule.
The sample evidence indicates the mean drivingdistance of Par, Inc. golf balls is greater than the meandriving distance of Rap, Ltd. golf balls.
17 © 2009 Econ-2030(Dr. Tadesse)
Using the p –Value Approach
4. Compute the p–value.For z = 6.49, the p –value < .0001.
Hypothesis Tests About m 1 m 2:s 1 and s 2 Known
5. Determine whether to reject H0.Because p–value < = .01, we reject H0. At the .01 level of significance, the sample evidenceindicates the mean driving distance of Par, Inc. golfballs is greater than the mean driving distance of Rap,Ltd. golf balls.
18 © 2009 Econ-2030(Dr. Tadesse)
Inferences About the Difference BetweenTwo Population Means: s 1 and s 2 Unknown
Interval Estimation of m 1 – m 2
Hypothesis Tests About m 1 – m 2
19 © 2009 Econ-2030(Dr. Tadesse)
Interval Estimation of m1 - m2:s 1 and s 2 Unknown
When s 1 and s 2 are unknown, we will:
• replace z/2 with t/2.
• we use the sample standard deviations s1 and s2
as estimates of s 1 and s 2 , and
20 © 2009 Econ-2030(Dr. Tadesse)
2 21 2
1 2 / 21 2
s sx x tn n
Where the degrees of freedom for t/2 are:
Interval Estimation of m1 - m2:s 1 and s 2 Unknown
Interval Estimate
22 21 2
1 22 22 2
1 2
1 1 2 2
1 11 1
s sn n
dfs s
n n n n
21 © 2009 Econ-2030(Dr. Tadesse)
Example
Difference Between Two Population Means:
s 1 and s 2 Unknown
Specific Motors of Detroithas developed a new automobileknown as the M car. 24 M carsand 28 J cars (from Japan) were roadtested to compare miles-per-gallon (mpg) performance. The sample statistics are shown on the next slide.
22 © 2009 Econ-2030(Dr. Tadesse)
Difference Between Two Population Means:
s 1 and s 2 Unknown
Sample Size
Sample MeanSample Std. Dev.
Sample #1M Cars
Sample #2J Cars
24 cars 28 cars
29.8 mpg 27.3 mpg2.56 mpg 1.81 mpg
23 © 2009 Econ-2030(Dr. Tadesse)
Difference Between Two Population Means:
s 1 and s 2 Unknown
Develop a 90% confidenceinterval estimate of the differencebetween the mpg performances ofthe two models of automobile.
24 © 2009 Econ-2030(Dr. Tadesse)
Point estimate of m1 m2 =x x1 2
Point Estimate of m 1 m 2
where:m1 = mean miles-per-gallon for the population of M carsm2 = mean miles-per-gallon for the population of J cars
= 29.8 - 27.3= 2.5 mpg
25 © 2009 Econ-2030(Dr. Tadesse)
Interval Estimation of m 1 m 2:s 1 and s 2 Unknown
The degrees of freedom for t/2 are:22 2
2 22 2
(2.56) (1.81)24 28
24.07 241 (2.56) 1 (1.81)
24 1 24 28 1 28
df
With /2 = .05 and df = 24, t/2 = 1.711
26 © 2009 Econ-2030(Dr. Tadesse)
Interval Estimation of m 1 m 2:s 1 and s 2 Unknown
2 2 2 21 2
1 2 / 21 2
(2.56) (1.81) 29.8 27.3 1.71124 28
s sx x tn n
We are 90% confident that the difference betweenthe miles-per-gallon performances of M cars and J carsis 1.431 to 3.569 mpg.
2.5 + 1.069 or 1.431 to 3.569 mpg
27 © 2009 Econ-2030(Dr. Tadesse)
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
Hypothesis Testing
1 2 02 21 2
1 2
( )x x Dts sn n
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 0 1 2 0: H D
m m 1 2 0: aH Dm m 0 1 2 0: H Dm m 1 2 0: aH D
Left-tailed Right-tailed Two-tailed Test Statistic
28 © 2009 Econ-2030(Dr. Tadesse)
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
Can we conclude, using a .05 level of significance, that themiles-per-gallon (mpg) performanceof M cars is greater than the miles-per-gallon performance of J cars?
29 © 2009 Econ-2030(Dr. Tadesse)
H0: m1 - m2 = 0
Ha: m1 - m2 > 0where: m1 = mean mpg for the population of M cars m2 = mean mpg for the population of J cars
1. Develop the hypotheses.
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
30 © 2009 Econ-2030(Dr. Tadesse)
2. Specify the level of significance.
3. Compute the value of the test statistic.
= .05
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
1 2 02 2 2 21 2
1 2
( ) (29.8 27.3) 0 4.003(2.56) (1.81)
24 28
x x Dts sn n
31 © 2009 Econ-2030(Dr. Tadesse)
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
Using the p –Value Approach4. Compute the p –value.
Compute the the degrees of freedom:22 2
2 22 2
(2.56) (1.81)24 28
24.07 241 (2.56) 1 (1.81)
24 1 24 28 1 28
df
32 © 2009 Econ-2030(Dr. Tadesse)
4. Determine the critical value and rejection rule.
Using the Critical Value Approach
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
For = .05 and df = 24, t.05 = 1.711
33 © 2009 Econ-2030(Dr. Tadesse)
Using the Critical Value Approach
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
5. Compare the Test Statistic with the Critical Value.
As t he test statistic 4.003 is greater than the Critical value 1.711, we reject H0.
We are at least 95% confident that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars?.
34 © 2009 Econ-2030(Dr. Tadesse)
5. Compute the P-value and determine whether to reject H0.
We are at least 95% confident that the miles-per-gallon (mpg) performance of M cars is greater than the miles-per-gallon performance of J cars?.
As the p–value < = .05, we reject H0.
Hypothesis Tests About m 1 m 2:s 1 and s 2 Unknown
Using the p-Value Approach
35 © 2009 Econ-2030(Dr. Tadesse)
In a matched-sample design each sampled item provides a pair of data values.
This design often leads to a smaller sampling error
than the independent-sample design because variation between sampled items is eliminated as a source of sampling error.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
36 © 2009 Econ-2030(Dr. Tadesse)
Example: A Chicago-based firm hasdocuments that must be quicklydistributed to district officesthroughout the U.S. The firmmust decide between two deliveryservices, UPX (United Parcel Express) and INTEX(International Express), to transport its documents.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
37 © 2009 Econ-2030(Dr. Tadesse)
In testing the delivery timesof the two services, the firm senttwo reports to a random sampleof its district offices with onereport carried by UPX and theother report carried by INTEX. Do the data on thenext slide indicate a difference in mean deliverytimes for the two services? Use a .05 level ofsignificance.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
38 © 2009 Econ-2030(Dr. Tadesse)
3230191615181410 716
25241515131515 8 911
UPX INTEX DifferenceDistrict OfficeSeattleLos AngelesBostonClevelandNew YorkHoustonAtlantaSt. LouisMilwaukeeDenver
Delivery Time (Hours)
7 6 4 1 2 3 -1 2 -2 5
Inferences About the Difference BetweenTwo Population Means: Matched Samples
39 © 2009 Econ-2030(Dr. Tadesse)
H0: md = 0
Ha: md Let md = the mean of the difference values for the two delivery services for the population of district offices
1. Develop the hypotheses.
Inferences About the Difference BetweenTwo Population Means: Matched Samples
40 © 2009 Econ-2030(Dr. Tadesse)
2. Specify the level of significance. = .05
3. Compute the value of the test statistic.
d dni ( ... ) .7 6 5
102 7
s d dndi
( ) . .2
176 1
92 9
2.7 0 2.942.9 10d
d
dts n
m
Inferences About the Difference BetweenTwo Population Means: Matched Samples
41 © 2009 Econ-2030(Dr. Tadesse)
4. Determine the critical value and rejection rule. Using the Critical Value Approach
For = .05 and df = 9, t.025 = 2.262.
5. Compare the Test Statistic with the Critical ValueBecause t = 2.94 > 2.262, we reject H0.
We are at least 95% confident that there is a difference in mean delivery times for the two services?
Inferences About the Difference BetweenTwo Population Means: Matched Samples
42 © 2009 Econ-2030(Dr. Tadesse)
5. Determine whether to reject H0.
We are at least 95% confident that there is a difference in mean delivery times for the two services?
4. Compute the p –value. For t = 2.94 and df = 9, the p–value is between.02 and .01. (This is a two-tailed test, so we double the upper-tail areas of .01 and .005.)
Because p–value < = .05, we reject H0.
Using the p –Value Approach
Inferences About the Difference BetweenTwo Population Means: Matched Samples